In this paper, we consider four common types of ruin probabilities for a discrete‐time multivariate risk model, where the insurer is assumed to be exposed to a vector of net losses resulting from a number of business lines over each period. By assuming a large initial capital for the risk model and regularly varying distributions for the net losses, we establish some interesting asymptotic estimates for ruin probabilities in terms of the upper tail dependence function of the net loss vector. Our results insightfully characterize how the dependence structure among the individual net losses affect the ruin probabilities in an asymptotic sense, and more importantly, from our main results, explicit asymptotic estimates for those ruin probabilities can be obtained via specifying a copula for the net loss vectors. Copyright © 2013 John Wiley & Sons, Ltd.